Introduction to Graphing: Solutions
Here are solutions to the questions on the worksheet accompanying the section Introduction to Graphing.
- What is the name of the function that is initially being
graphed?
- Answer
- Looking at the list of functions on the left, we see that
"abs" is selected. This looks like the absolute
value function.
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- What is the "plot-window" (i.e., the limits on
the horizontal and vertical axes) for this plot?
- Answer
- Looking at the control boxes on the right, or by estimating from
the graph itself, we see that the plot only shows the values between
-5 and 5 on both the horizontal and vertical axes..
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- Describe precisely what happens to limits on the "plot-window" when
you hit the "Zoom In", "Zoom Out", and
"Restore" buttons?
- Answer
- In this case, "Zoom In" divides the limits by 2, while "Zoom Out"
multiplies the limits by 2. That is because the graph happens
to be centered on the origin, (0, 0). In general, the center
point of graph is kept the same, while the size of the horizontal
and vertical range is divided or doubled. In any case, the
"Restore" button returns the limits to their original
settings.
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- What are the eight commands listed in the top menu?
- Answer
- Clicking on the menu reveals the commands:
- Main Screen
- Multigraph Utility
- Animate
Utility
- Parametric Curves Utility
- Derivatives
Utility
- Riemann Sums Utility
- Integral Curves
Utility
- Graph 3D Utility
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- What value does the program give for the absolute value of
-3?
- Answer
- The program displays the equation abs(-3) = 3. That is, the
absolute value function takes an input of -3 to an output of
3. We normally write this as |-3| = 3. This corresponds
to the point on the graph (-3, 3).
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- Give a rule, in words, that describes the floor
function.
- Answer
- Looking at the results:
| x |
floor(x) |
-2.5
3
4.12 |
-3
3
4 |
- It rounds down to the next lower integer.
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- Describe all the changes you observe in the "Main
Screen" window.
- Answer
- In the list of functions on the left, we can see that our new
function, f, now appears. The
plot of f also appears as a fairly
steep parabola opening upwards.
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- From the graph, estimate the domain
and range of g.
- Answer
- The graph seems to go off to positive "infinity"
vertically, as it goes to negative infinity horizontally. It
also appears that the function decreases in value, as we move from
left to right, ending at the point (3, -5). This means that
the domain must be (-¥, 3] (i.e., all x
values less than or equal to 3) and the range is [-5, ¥)
(i.e., all y values greater than or equal to -5).
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- Use the "x =" box to calculate h(-2),
h(-1), h(0), h(1),
h(2), and h(3).
Note: This is very similar to what we did by hand in a previous
Exercise; in this way, you will be able to use XFunctions to create your
own Exercises and check your own work.
- Answer
- Entering each value, in turn, gives the following table of values:
| x |
h(x) |
-2
-1
0
1
2
3 |
12
-3
-1
1
undefined
undefined |
- which corresponds to the formula:
-
- since this rule does not include values for x
= 2 or x = 3. Notice how
it gives a value for x = -2, even
though the y-value is too large for it to appear inside the
plot window.
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- Use the "x =" box to calculate k(-2),
k(-1), k(-0.5), k(1.5),
and k(3). Notice how it actually extends
the domain of the function to include values that are not in our original
table, but not over the entire real line.
- Answer
- Entering each value, in turn, gives the following table of values:
| x |
k(x) |
-2
-1
-0.5
1.5
3 |
1
-2
-1.5
2
undefined |
- Notice how, by specifying the function to be "piecewise
linear", it connected-the-dots to extend the domain to all
values between -2 and 2, so we obtain values for points that were
not in the original table. However, it does not gives a
value for x = 3, since that is outside
the range of values in the table.
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- Use the "x =" box to calculate q(-2),
q(-1), q(-0.5), q(1.5), and
q(3). Compare these values with those of k.
- Answer
- Entering each value, in turn, gives the following table of
approximate values:
| x |
q(x) |
-2
-1
-0.5
1.5
3 |
1
-2
-2.123
3.931
undefined |
- At x = -2, -1, 0, 1, 2, we obtain the
same values as k, since that is how we
constructed the graph. However, this time the function is
extended in very complex way to all other values between -2 and
2. Again, XFunctions will not give a value for x
= 3, since it is outside the specified domain of the function
(i.e., the established horizontal limits of the graph).
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- Compare the graphs of the two functions algebraically
(i.e., the two formulas) and geometrically. Try to relate the two
comparisons.
- Answer
- In mathematical notation, the functions are y = ëxû
and y = 2ëxû,
that is, the formula for the second function is simply 2 times that
of the first. This means that, for any given x-value,
the y-value of the second is always twice that of the
first. The second graph appears to be "steeper" than
the first, because all of the "steps" are twice (i.e., 2 times) as high.
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- Compare the graphs of y = ëxû
and y = -ëxû
algebraically and geometrically
(remember this is just mathematical notation for the two functions you have
just plotted). Try to relate the two
comparisons.
- Answer
- This time, the formula for the second function is simply negative
that
of the first. This means that, for any given x-value,
the y-value of the second is always negative that of the
first. The second graph appears to be the same as the first,
but flipped over. We know that multiplying by -1 flips
over the number line; since we are multiplying y-values, we
can imagine this graph as the result of flipping the vertical
axis.
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Go to Arithmetic and Graphing.